Yu Li1, Jie Liu1, Yida Deng1, Xiaopeng Han1, Wenbin Hu1, Cheng Zhong1. 1. Key Laboratory of Advanced Ceramics and Machining Technology (Ministry of Education), School of Materials Science and Engineering and Tianjin Key Laboratory of Composite and Functional Materials, School of Materials Science and Engineering, Tianjin University, No. 135 Yaguan Road, Jinnan District, Tianjin 300350, China.
Abstract
With the aid of X-ray microcomputed tomography and digital image processing technology, the three-dimensional structure of foam zinc prepared by the electrodeposition process is reconstructed. Furthermore, a simplified finite-element model of foam zinc, which can more accurately reflect its structure, is proposed. Based on the Brinkman-Forchheimer-extended-Darcy law, the finite-element method is used for the numerical simulation of the percolation performance of the foam zinc. The results indicate that for high-porosity foam zinc, the pore density is the main factor affecting its percolation performance. A function is established to describe the relationship between the pore density and pressure drop. To obtain an optimum structure, a tetrakaidecahedron cylinder model is established and compared to a previously built model, and the comparison demonstrates that the optimized model performs better in the field of percolation performance.
With the aid of X-ray microcomputed tomography and digital image processing technology, the three-dimensional structure of foam zinc prepared by the electrodeposition process is reconstructed. Furthermore, a simplified finite-element model of foam zinc, which can more accurately reflect its structure, is proposed. Based on the Brinkman-Forchheimer-extended-Darcy law, the finite-element method is used for the numerical simulation of the percolation performance of the foam zinc. The results indicate that for high-porosity foam zinc, the pore density is the main factor affecting its percolation performance. A function is established to describe the relationship between the pore density and pressure drop. To obtain an optimum structure, a tetrakaidecahedron cylinder model is established and compared to a previously built model, and the comparison demonstrates that the optimized model performs better in the field of percolation performance.
The rechargeable metal–air
battery has advantages such as
low cost, environmental friendliness, and high safety performance;
especially, zinc–air batteries are expected to be one of the
most promising new energy batteries for the next generation because
of their high energy density.[1−4] The theoretical energy density can reach as high
as 1350 kW h kg–1.[5−7] However, because of technical
limitations, the rechargeable zinc–air batteries have not yet
been commercialized.[8−10] Taking the zinc anode as an example, it is much denser
and has a smaller specific surface area; dendritic or mossy growth
of zinc results in the morphology and shape change during the charging/discharging
process, leading to the decline in the performance and the low cycling
stability of the battery.[11,12]Because of its
three-dimensional (3D) through-pore structure, foam
metal has a low density, high specific surface area, low flow resistance,
good mechanical properties, and good thermal management performance.[13,14] As a structural and functional integrated material,[15,16] foam metal can be used as an electrode material for batteries. Foam
Zn electrode with a higher specific surface area increases the charge/discharge
capacity and enhances the rated performance of batteries; therefore,
the charge/discharge efficiency of the battery is improved and the
energy loss is reduced. The pores can also provide sufficient space
for the growth of dendrites formed by the Zn electrode during the
charging/discharging process, improving the performance of the Zn
electrode and providing the flow channel of the electrolyte. The good
performance of the foamed structure on the thermal management can
improve the heat dissipation efficiency and the thermal failure of
materials or structures resulting from the heat concentration during
the battery charging and discharging process.[17] The high porosity of the porous structure can greatly reduce the
density of the electrode, reducing the weight of the battery to increase
the current density per unit mass, and promoting the development of
low-weight and heterotypic battery cells. The researchers studied
the percolation properties of foam metal with gas and liquid as working
fluids.[18,19] It was found that the increase in pressure
drop is exponentially related to pore density.[20] For foam metals with a high porosity, a large number of
irregular zigzag flow channels can continuously disturb the fluid
boundary layer.[21] Fluid flow often occurs
as reflux, turbulence, and unsteady flow, making the influencing factors
of seepage characteristics extremely complex.[22,23] The performance analysis of porous materials has been carried out
using simplified mathematical models.[24,25] Although these
models are quite different from the actual structure of foam metal,
they are relatively simple and easy to be used for experimental analysis,
and the simulation results are consistent with most experimental results.[26,27] The applications of the finite-element method in numerical simulation
of the percolation properties of foam metal is of great practical
significance.[28,29] Studies on percolation properties
of foam zinc contribute to comprehend the flow behavior of electrolyte
in foam zinc electrode, which could provide a theoretical basis for
the selection of the pore structure parameters of foam zinc as the
anode of the zinc–air battery regarding the flow resistance.At first, foam zinc as the research object in this study is prepared
by the ultrasonic-assisted electrodeposition process.[30] The 3D structure of the foam zinc is reconstructed. Furthermore,
a simplified finite-element model of the foam zinc, which can reflect
its actual structure, is proposed. Based on the Brinkman–Forchheimer-extended-Darcy
law, the finite-element method[31,32] is applied in the finite-element
analysis (FEA) for revealing the quantitative relationship between
the pore structure and the percolation performance of foam zinc, and
the foam zinc structure is optimized. Through the numerical simulation
of the related process, a continuous and complete performance curve
of the pressure drop and flow velocity is obtained,[23,33] which provides the theoretical basis for the selection of the pore
structure parameters of foam zinc as the anode of the zinc–air
battery and shortens the test period.
Results
and Discussion
The average pore diameter dp of the
foam zinc samples obtained by the electrodeposition process is measured
using the direct section observation method. Based on this, foam zinc
cell models with different parameters were established. According
to the pore density, it can be divided into four types: 10, 25, 45,
and 70 PPI and 90, 93, 95, and 97%, respectively, according to porosity.
The pore structure parameters are presented in Table . There are about 700 000 nodes and
340 000 elements in the tetrakaidecahedron tri-prism cell model.
Table 1
Pore Structure Parameters
dp (mm)
pore density (PPI)
porosity (%)
5.35
10
90
5.35
10
93
5.35
10
95
5.35
10
97
2.16
25
97
1.14
45
97
0.67
70
97
Validation
of FEA
The finite-element
model of foam zinc with a pore density of 10 PPI and porosity of 93%
is selected, and the pressure drop of foam zinc per unit length at
different velocities is calculated; the results of the comparison
are presented in Figure . Figure presents
the FEA results compared to the experimental results using water as
the fluid.
Figure 1
FEA results compared to the experimental results using water as
the fluid.
FEA results compared to the experimental results using water as
the fluid.By comparison, it is found that
the FEA results agree well with
the experimental results and the data available in the literature.[17,34] It is revealed that the model and finite-element method established
in this paper can be applied to the research on the percolation performance
of foam metal.
Influence of Porosity on
the Percolation Performance
of Foam Zinc
Foam zinc samples with a pore density of 10
PPI and porosity varying from 90 to 97% are selected as the objects
to study the influence of porosity on the percolation properties of
foam metal. A part of the FEA is presented in Figure . Figure indicates that with the flow rate v increasing from 0.4 to 1.2 m s–1, the unit pressure
drop ΔP/L increases continuously,
and the growth rate of the unit pressure drop ΔP/L increases with the flow rate v. Under the same flow rate, the unit pressure drop ΔP/L increases continuously with the decrease
in porosity. The results indicate that it will lead to an increase
in the fluid flow resistance with the increase in flow velocity v or decrease in porosity. It is known from the curve that
the unit pressure drop ΔP/L does not conform to a linear relationship with the flow velocity v, indicating that the flow state of the water in the foam
zinc with a pore density of 10 PPI and porosity of 90–97% deviates
from the laminar state described by Darcy’s law (5).
Figure 2
Pressure distributions of foam zinc with a pore density of 10 PPI
and different porosities at flow velocity v = 0.8
m s–1 ((a) 10 PPI, ε = 90; (b) 10 PPI, ε
= 97).
Pressure distributions of foam zinc with a pore density of 10 PPI
and different porosities at flow velocity v = 0.8
m s–1 ((a) 10 PPI, ε = 90; (b) 10 PPI, ε
= 97).The pressure difference of unit
length ΔP/L under different
velocities v was calculated, and the relationship
between these two parameters
is illustrated in Figure .
Figure 3
Dependence of ΔP/L on v for foam zinc with a pore density of 10 PPI.
Dependence of ΔP/L on v for foam zinc with a pore density of 10 PPI.The FEA results obtained in Figure are processed to obtain the relationship
curve between
ΔP/Lv and the velocity v, as shown in Figure . The curve indicates that there is a linear relationship
between ΔP/Lv and the flow
velocity v, that is to say, the unit distance pressure
difference ΔP/L and velocity v exhibit a quadratic function relation, which is in agreement
with the Brinkman–Forchheimer-extended-Darcy law (6), indicating that the flow state of the fluid in foam zinc
with a pore density of 10 PPI and porosity of 90–97% is a laminar
turbulent complex state. Under this condition, the flow resistance
of foam zinc is affected by the laminar flow and turbulent flow of
the fluid.
Figure 4
Dependence of ΔP/Lv on v for foam zinc with a pore density of 10 PPI.
Dependence of ΔP/Lv on v for foam zinc with a pore density of 10 PPI.The percolation performance of foam metals can
be characterized
by the viscosity percolation performance coefficient k1 and the inertial percolation performance coefficient k2. The larger k1 and k2 are, the better the percolation
performance of the foam metal. The k1 and k2 of foam zinc with a pore density of 10 PPI
and porosity of 90–97% can be calculated by the Brinkman–Forchheimer-extended-Darcy
law (6) and the relationship curve between ΔP/Lv and v in Figure .Table indicates
that when the pore density of the foam metal is constant, the skeleton
of foam metal becomes thinner with the increase in porosity, the effective
cross-sectional area of the flow increases, and the resistance to
fluid flow decreases. At the same time, as the skeleton of the foam
metal gets disturbed, the disturbance of the foam metal skeleton decreases
with the porosity of the foam metal, which is beneficial to the fluid
flow in that it improves the percolation performance of foam metal.
Table 2
Viscosity Percolation Performance
Coefficient k1 and Inertial Percolation
Performance Coefficient k2 of Foamed Zinc
with Different Porosities
k1 × 107 (m2)
k2 × 103 (m)
10 PPI, ε = 90
1.87
2.13
10 PPI, ε = 93
2.21
3.08
10 PPI, ε = 95
2.39
4.32
10 PPI, ε = 97
2.94
6.77
Influence of Pore Density
on the Percolation
Performance of Foam Zinc
Foam zinc samples with a porosity
of 97% and pore density varying from 10 to 70 PPI are selected as
the objects to study the influence of the pore density on the percolation
properties of foam metal. A part of the FEA is shown in Figure .
Figure 5
Press distributions of
foam zinc with a pore density of 10 PPI
and different porosities at flow velocity v = 0.8
m s–1 ((a) 25 PPI, ε = 97; (b) 70 PPI, ε
= 97).
Press distributions of
foam zinc with a pore density of 10 PPI
and different porosities at flow velocity v = 0.8
m s–1 ((a) 25 PPI, ε = 97; (b) 70 PPI, ε
= 97).The relationship curves between
the ΔP/Lv and velocity v, as shown in Figure , are obtained from
the FEA results above, and the k1 and k2 of foam zinc with different pore densities
are listed in Table .
Figure 6
Dependence of ΔP/Lv on v.
Table 3
Viscosity
Percolation Performance
Coefficient k1 and Inertial Percolation
Performance Coefficient k2 of Foamed Zinc
with Different Porosities
k1 × 107 (m2)
k2 × 103 (m)
10 PPI, ε = 90
29.40
6.77
10 PPI, ε = 93
5.09
2.90
10 PPI, ε = 95
1.73
1.51
10 PPI, ε = 97
0.66
0.84
Dependence of ΔP/Lv on v.Table indicates
that as the pore density of foam zinc increases from 10 to 70 PPI, k1 decreases from 29.40 × 10–8 to 0.66 × 10–8 m2 and k2 decreases from 6.77 × 10–3 to 0.84 × 10–3 m, which reveals that although
the porosity of foam zinc is constant, the low pore density will result
in a lower flow resistance and better percolation performance. Comparing
the ranges of k1 and k2 in Table with those in Table , it is known that under high-porosity (greater than 90%) conditions,
the pore density is the main variable that affects the percolation
performance.The Ergun model is an empirical equation based
on the pore structure
to describe the flow resistance of porous materials. The Ergun-like
model has been proposed for foam metals by Dukhan and others.[34−36]Based on the Ergun-like model, the influence
of aperture (pore density) on the percolation performance is investigated
in this study. Formula can be written ask1 and k2 can be obtained by combining with formula The
relationships between k1, k2, and the aperture dp can
be obtained from the fitting formulas and 4 with the data of both dp in Table and k1 and k2 in Table The formula
of the flow resistance of the
foam zinc with a porosity of 97% is obtained by substituting formulas and 6 into the Forchheimer-extended-Darcy law expression (0.67 mm ≤ dp ≤
5.35 mm, 0.4 m s–1 ≤ v ≤
1.2 m s–1)
Optimum
Structural Design of Foam Zinc
The framework of the FEA model,
namely, the tetrakaidecahedron
tri-prism model of foam zinc based on the real structure, was optimized.
A tetrakaidecahedron cylinder model of foam zinc is established, as
shown in Figure .
Figure 7
Tetrakaidecahedron
cylinder model ((a) tetrakaidecahedron cylinder
model; (b) tetrakaidecahedron cylinder cell model and cross section
of prism).
Tetrakaidecahedron
cylinder model ((a) tetrakaidecahedron cylinder
model; (b) tetrakaidecahedron cylinder cell model and cross section
of prism).The two models with the same pore
structure parameters (pore density
of 10 PPI and porosity of 97%) are selected for FEA calculation. The
definitions of the boundary condition and application of the load
are identical. The FEA results of the two different models are presented
in Figures and 9.
Figure 8
Press distributions with a pore density of 10 PPI at flow
velocity
of v = 0.8 m s–1 ((a) tetrakaidecahedron
tri-prism model; (b) tetrakaidecahedron cylinder model).
Figure 9
FEA results ((a) dependence of ΔP/L on v; (b) dependence of ΔP/Lv on v).
Press distributions with a pore density of 10 PPI at flow
velocity
of v = 0.8 m s–1 ((a) tetrakaidecahedron
tri-prism model; (b) tetrakaidecahedron cylinder model).FEA results ((a) dependence of ΔP/L on v; (b) dependence of ΔP/Lv on v).Figures and 9 indicate that with the increase
in flow velocity v, the unit pressure drop ΔP/L of the two structures both increases.
When the flow velocity v is constant, the ΔP/L of the tetrakaidecahedron cylinder
model is smaller, that is to
say, it has a lower flow resistance. The ΔP/Lv of the two models has a linear relationship
with the flow velocity v. The percolation factors
of the two models are obtained using a linear fitting method and presented
in Table .
Table 4
Viscosity Percolation Performance
Coefficient k1 and Inertial Percolation
Performance Coefficient k2 of Different
Models
tetrakaidecahedron tri-prism model
tetrakaidecahedron cylinder model
k1 × 107 (m2)
2.94
1.69
k2 × 103 (m)
6.77
17.76
Table indicates
that the tetrakaidecahedron cylinder model has a smaller viscous seepage
coefficient and a lower flow resistance but a greater inertial percolation
performance coefficient. When the flow velocity v is increased to a certain extent, the inertial effect gradually
replaces the viscous effect as the leading factor affecting the flow;
then, the tetrakaidecahedron cylinder model will perform well regarding
percolation.
Conclusions
In this
study, we have prepared the open-cell foam zinc by the
ultrasonic-assisted electrodeposition process. The tetrakaidecahedron
tri-prism model is established based on the real structure using X-ray
micro-CT as the FEA model of foam zinc. Furthermore, FEA calculations
of the percolation performance of foam zinc are carried out, which
indicated that pore density is the main factor affecting the percolation
performance of foam zinc. According to the FEA results, the empirical
relationship between the fluid flow resistance in foam zinc and the
aperture is obtained.Formula can provide
a theoretical basis for the selection
of the pore structure parameters of foam zinc as the anode of the
zinc–air battery regarding the flow resistance.The tetrakaidecahedron
cylinder model of foam zinc is established
by optimizing the tetrakaidecahedron tri-prism model. By comparing
the percolation performance of the two models, it can be concluded
that the tetrakaidecahedron cylinder model will perform well regarding
percolation. The optimized 3D network structure can provide theoretical
guidance for the preparation of high-performance foam zinc.In the process of building up the finite-element models and setting
the boundary conditions, a reasonable degree of simplification is
made in both mathematics and physics. Although the mathematical and
physical simplifications are carried out in this article, the conclusions
obtained are instructive on predicting the percolation properties
of foam zinc.
Experimental Section
Experimental Materials
Polyurethane
sponge of 10 PPI is used as the matrix, and foam zinc is prepared
by the electrodeposition process. The polyurethane foam is cut into
the specimen with a size of 140 mm × 110 mm × 15 mm. The
specimen is pretreated through the following steps: degreasing, roughening,
sensitizing by SnCl4 and PdCl2, and activating
and then peptizing before plating. Chemical zinc-plating technology
with the assistance of ultrasound is adopted to perform the conductivity
treatment. The plating solution is developed according to the formula
of Table and then
put into the ultrasonic generator. When the ultrasonic power is adjusted
to 300 W and the current is adjusted to 12 A, the specimen is immersed
in the plating solution. After undergoing electrodeposition for 10
h, the specimen is taken out, washed by clear water, and finally dried
in an oven. The foam zinc is processed into the specimen with a size
of 60 mm × 40 mm × 10 mm using the wire electrode cutting
technology. Foam zinc is heat treated at 360 °C with hydrogen
atmosphere for 8 h, and then heat treated at 100–150 °C
with hydrogen atmosphere for 8 h to remove polyurethane sponge. Table lists the components
of the electroplating solution.
Table 5
Components of Electroplating
Solution
component
concentration
ratio
ZnSO4·7H2O (AR)
12–23 g L–1
EDTA·2Na (AR)
15–20 g L–1
HCHO
(36%) (AR)
10–15 mL L–1
KNaC4H4O6·4H2O (AR)
relative
K4Fe(CN)6 (AR)
relative
Calculation Method of Percolation
Performance
Assuming the fluid is turbulent in foam zinc
without a phase transition,
the two-equation k–ε turbulence model[37] is used as followswhereandThe pressure
gradient of ΔP/L of the fluid
in the porous media is linear with
velocity v̅ under a low flow velocityThe viscous percolation performance coefficient k1 of the foam metal can be calculated using formula .When the
flow rate rises, the relationship in the porous media between the
pressure gradient ΔP/L and
velocity v̅ deviates from the Darcy’s
law, where the Brinkman–Forchheimer-extended-Darcy law is satisfiedThe inertial percolation performance coefficient k2 of foam metal can be calculated using formula .Both k1 and k2 are the intrinsic
parameters related to the pore structure
characteristics of the foam metal, which do not change with the fluid
properties. Therefore, these two parameters can be used to describe
the percolation performance of the foam metal.The device for
testing the percolation performance of the foam
zinc is illustrated in Figure . During the experiment, we first investigate the sealing
performance of the test device and then open the cooling cycle system
for cleaning and cooling using the cycle of deionized water in the
device. By adjusting the flow rate Qf,
the pressure difference ΔP of deionized water
flowing through the foam zinc samples with different pore densities
is measured at different flow rates v̅,[38] and k1 and k2 are calculated using the above-mentioned formulas.
Figure 10
Schematic
of testing equipment for percolation performance of foam
metal ((a) model diagram; (b) sketch map).
Schematic
of testing equipment for percolation performance of foam
metal ((a) model diagram; (b) sketch map).Using the Ansys Fluent software package, the standard two-equation k−ε turbulence model is selected. The material
property of the foam metal is set to foam zinc, the fluid is set to
water,[39] and the inlet and outlet of the
flow channel are, respectively, set to the speed inlet and the pressure
outlet. The pore diameter is used as the length scale in the FEA.[40]
Finite-Element Model of
Foam Zinc
The percolation performance test does not destroy
the structure of
the foam zinc, and the test specimen is used for the microcomputed
tomography (micro-CT) scanning.[41] The foam
zinc structure is scanned using a Bruker Micro-CT SKYSCAN 1172 (made
in Belgium). The acceleration voltage and current of the X-rays are
set to 100 kV and 100 μA, respectively. During the experiment,
although the increment of the fixed angle is set to 0.7°, we
select the appropriate scanning resolution of 12 μm for the
pore density of 10 PPI and finally produce a set of two-dimensional
(2D) tomography images (Figure a) of the foam zinc specimens. Using the extreme point
threshold method, the approximate range of the gray threshold is determined
within the values ranging from 4 to 20 (Figure b). The image segmentation is processed
in MATLAB, the statistical data of surface density are calculated,
and the porosity of foam zinc with a certain threshold is calculated
using the Origin software (shown in Table ). Compared with the porosity measured by
experiments, the final threshold value of the specimens is determined
to be 12. Then, the 3D structure of the foam zinc is reconstructed.[42,43]Figure c presents
the full structure, whereas Figure d,e presents the partial structures of the foam zinc.
It can be drawn from Figure c,e that the internal spatial structure of foam zinc is regular
with its holes evenly distributed and isotropic. The 3D structure
of pores in the foam zinc is approximately tetrakaidecahedral, whereas
the 2D structures are roughly quadrilateral and hexagonal. The tetrakaidecahedron
tri-prism model is proposed by cutting off prisms inside the tetrakaidecahedra
by the pretreatment module of the ANSYS software. Considering the
geometric characteristics of the tetrakaidecahedron, the spatial topology
of the original models is based on the close-packed structure; then,
we perform Boolean intersection operation with a cylinder. Finally,
a simplified finite-element model of foam zinc is proposed, as shown
in Figure f. Figure g shows the tetrakaidecahedral
tri-prism cell model, which can accurately reflect the structure of
foam zinc.
Figure 11
Procedure of tetrakaidecahedron tri-prism model establishment
((a)
2D tomography image; (b) grayscale maps; (c) full structures; (d,
e) partial structure; (f) tetrakaidecahedron tri-prism cell model;
(g) tetrakaidecahedron tri-prism model).
Table 6
Porosity of Different Thresholds and
Experiments
threshold value (T)
porosity (%)
8
94.313
10
94.301
12
94.284
14
94.265
experimental
value
94.28
Procedure of tetrakaidecahedron tri-prism model establishment
((a)
2D tomography image; (b) grayscale maps; (c) full structures; (d,
e) partial structure; (f) tetrakaidecahedron tri-prism cell model;
(g) tetrakaidecahedron tri-prism model).
Authors: Katherine J Harry; Daniel T Hallinan; Dilworth Y Parkinson; Alastair A MacDowell; Nitash P Balsara Journal: Nat Mater Date: 2013-11-24 Impact factor: 43.841
Authors: Kyung E K Sun; Tuan K A Hoang; The Nam Long Doan; Yan Yu; Xiao Zhu; Ye Tian; P Chen Journal: ACS Appl Mater Interfaces Date: 2017-03-08 Impact factor: 9.229
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